CN110034746A - One kind is based on maximum collaboration entropy kalman filter method - Google Patents

Abstract

The embodiment of the invention discloses one kind based on maximum collaboration entropy kalman filter method, there is very strong robustness to pulsed non-Gaussian noise, and the state mean value communication process of traditional Kalman filter algorithm is maintained, and remain the communication process of the matrix of predicting covariance.Therefore, this new filter also has recursive structure, is suitable for online updating.

Description

Kalman filtering method based on maximum collaborative entropy

Technical Field

The invention relates to the field of intelligent control, in particular to a Kalman filtering method based on maximum collaborative entropy.

Background

The conventional kalman filter is based on the minimum mean square error, which performs well under gaussian noise. However, many practical engineering scenarios cannot satisfy the assumption of gaussian noise, which results in the performance degradation of the conventional kalman filter in the application of non-gaussian noise interference, especially impulse-type non-gaussian noise interference, and the main reason is that the conventional kalman filter algorithm can only give reliable estimation when the noise is gaussian distributed, and when the conventional kalman filter is applied to the non-gaussian condition, the performance of the conventional kalman filter may be degraded, especially when the system is interfered by impulse noise. Impulse noise has a heavy tail distribution (e.g., some mixed gaussian distributions), which is common in many real automatic control and target tracking scenarios. The main reason for this problem is that the conventional kalman filter is based on the minimum mean square error criterion, which is very sensitive to large outliers, resulting in a deterioration of the robustness of the conventional kalman filter in non-gaussian noise environments.

Disclosure of Invention

The technical problem to be solved by the embodiment of the invention is to provide a Kalman filtering method based on maximum collaborative entropy. The method has strong robustness to impulse type non-Gaussian noise, keeps the state mean value propagation process of the traditional Kalman filtering algorithm, and keeps the propagation process of the matrix of the covariance of the prediction error.

In order to solve the above technical problem, an embodiment of the present invention provides a kalman filtering method based on maximum collaborative entropy, including the following steps:

s1 setting the kernel bandwidth sigma and a small positive number η, setting the initial state estimateAnd an initial covariance matrix P (0|0), t ═ 1;

s2: obtained using the following formulaAnd P (t | t-1), B is obtained by Cholesky decomposition_p(t|t-1)，

P(t|t-1)＝A(t-1)P(t-1|t-1)A^T(t-1)+Q

Wherein A (k-1) is a control parameter of the system,for the result of the prediction of the last state,is the optimum result of the last state, P (k | k-1) isCorresponding covariance, Q is the covariance of the system process;

s3: so that j is equal to 1, and the ratio is zero, wherein Representing the state estimate in the jth fixed point iteration;

s4: the posterior estimate is calculated using the following formula

wherein ：

wherein ,denotes the kalman gain at time t, H denotes the measurement matrix, z denotes the measurement value,representing the covariance of the measurement noise, pair E [ β (t) β^T(t)]Cholesky decomposition (square root method) was performed to obtain B,ξ (t) denotes measurement noise, symbol E denotes the desired operator, diag () denotes extraction of diagonal elements, G_σRepresenting a gaussian kernel. d_i(t) is the i-th element of D (t), w_i(t) is the i-th line element of W (t), where

S5: when the following formula holds, thenOtherwise, let j +1 → j, repeat S4,

s6: the a posteriori estimated covariance matrix is updated using the following formula, let t +1 → t, repeat S2,

the embodiment of the invention has the following beneficial effects: the method has strong robustness to impulse type non-Gaussian noise, maintains the state mean value propagation process of the traditional Kalman filtering algorithm, and reserves the propagation process of the matrix of the prediction error covariance. Therefore, the new filter also has a recursive structure, and is suitable for online updating.

Drawings

Fig. 1 is a schematic flow chart of a kalman filtering prediction and update method based on maximum collaborative entropy.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.

For the standard kalman filter algorithm, the noisy state model and measurement model are recorded as:

where A (k-1) is a control parameter of the system, here a matrix. H is a measurement matrix. The Kalman filtering model assumes that the true state at time t evolves from time (t-1). As in many practical cases, dynamic systems can only obtain input data with noise.

Assuming that the measurement set at time t is Z ═ { Z (t) }, the kalman filtering algorithm is two steps:

(1) the a priori estimate is expressed as

P(t|t-1)＝A(t-1)P(t-1|t-1)A^T(t-1)+Q#(3)

For the result of the prediction of the last state,is the result of the last state optimization. P (k | k-1) isThe corresponding covariance. Q is the covariance of the system process.

(2) The posterior estimate is

K(t)＝P(t|t-1)H^T(t)(H(t)P(t|t-1)H^T(t)+R)^-1#(4)

P_t＝(1-K(t)H(t))P(t|t-1)#(6)

K (t) represents the Kalman gain,i.e. the a posteriori state estimation, P_tIs updated for the error covariance matrix.

Due to the fact that non-Gaussian noise exists in traffic flow data, the performance of a traditional Kalman filtering model is deteriorated. Therefore, the embodiment of the invention provides an improved Kalman filtering applied to traffic flow prediction, and the embodiment is as follows.

For the state model and linear model described above, there are:

in this connection, it is possible to use,for E [ β (t) β (t)^T]Cholesky decomposition (square root method) is performed to obtain B (t), and symbol E represents the desired operator. Left-hand B of equation 7^-1(t) obtaining d (t) w (t) x (t) + e (t), wherein

e(t)＝B^-1(t)β(t)。

The cost function based on the maximum collaborative entropy is as follows:

where L denotes the dimension of D (t), d_i(t) denotes the i-th element of D (t), w_i(t) denotes the ith row of W (t). e.g. of the type_i(t) is the i-th element of e (t). Symbol G_σRepresenting a gaussian kernel, and σ is the gaussian kernel bandwidth. Under the maximum co-entropy criterion, the optimization estimates are as follows:

specifically, the present embodiment is mainly performed by the following steps.

(1) Selecting a suitable kernel bandwidth σ and a small positive number η, setting the initial state estimate

And an initial covariance matrix P (0|0), t ═ 1;

(2) obtained by equation 2 and equation 3And P (t | t-1), B is obtained by Cholesky decomposition_p(t|t-1)。

(3) So that j is equal to 1, and the ratio is zero, wherein Representing the state estimate in the jth fixed point iteration.

(4) Calculating the posterior estimates using equation 10, equation 11

wherein

wherein ,denotes the kalman gain at time t, H denotes the measurement matrix, z denotes the measurement value,representing the covariance of the measurement noise, pair E [ β (t) β^T(t)]Cholesky decomposition (square root method) was performed to obtain B,ξ (t) denotes measurementNoise, symbol E representing the desired operator; diag () denotes extracting the diagonal element, G_σRepresenting a gaussian kernel. d_i(t) is the i-th element of D (t), w_i(t) is the i-th line element of W (t), where

(5) If equation 12 holds, makeThen (6) is performed, otherwise, let j +1 → j, go back to (4).

(6)

(7) The a posteriori estimated covariance matrix is updated with equation 13, let t +1 → t, return to (2).

As a priori-a posteriori estimation algorithm, the kalman filter algorithm based on maximum co-entropy can be summarized as a prediction-update equation, as shown in fig. 1.

The embodiment of the invention has the following beneficial effects:

1. the invention designs a Kalman filtering algorithm based on maximum collaborative entropy, and the algorithm uses the maximum collaborative entropy criterion to replace the traditional least square error criterion as the optimal target for derivation, so that the interference of pulse type non-Gaussian noise can be well processed.

2. Because the optimal target criterion of the maximum collaborative entropy is not a convex function, the invention designs a new fixed-point algorithm to update the posterior estimation.

3. The method not only keeps the state mean value propagation process of the traditional Kalman filtering algorithm, but also keeps the propagation process of the matrix of the prediction error covariance. Therefore, the new filter also has a recursive structure, and is suitable for online updating.

While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.

Claims (1)

1. A Kalman filtering method based on maximum collaborative entropy is characterized by comprising the following steps:

s1 setting the kernel bandwidth sigma and a small positive number η, setting the initial state estimateAnd an initial covariance matrix P (0|0), t ═ 1;

s2: obtained using the following formulaAnd P (t | t-1), R being obtained by Cholesky decomposition_p(t|t-1)，

P(t|t-1)＝A(t-1)P(t-1|t-1)A^T(t-1)+Q

Wherein A (k-1) is a control parameter of the system,for the result of the prediction of the last state,is the optimum result of the last state, P (k | k-1) isCorresponding covariance, Q is the covariance of the system process;

s3: so that j is equal to 1, and the ratio is zero, wherein Representing the state estimate in the jth fixed point iteration;

s4: the posterior estimate is calculated using the following formula

wherein ：

wherein ,denotes the kalman gain at time t, H denotes the measurement matrix, z denotes the measurement value,representing the covariance of the measurement noise, pair E [ β (t) β^T(t)]Cholesky decomposition (square root method) was performed to obtain B,ξ (t) denotes measurement noise, symbol E denotes the desired operator, diag () denotes extraction of diagonal elements, G_σDenotes the Gaussian nucleus, d_i(t) is the i-th element of D (t), w_i(t) is the i-th line element of W (t), where

S5: when the following formula holds, thenOtherwise, let j +1 → j, repeat S4,

s6: the a posteriori estimated covariance matrix is updated using the following formula, let t +1 → t, repeat S2,